Chamfer (geometry)

In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another.

For a polyhedron, this operation adds a new hexagonal face in place of each original edge.

The chamfered tetrahedron or alternate truncated cube is a convex polyhedron constructed: For a certain depth of chamfering/truncation, all (final) edges of the cT have the same length; then, the hexagons are equilateral, but not regular.

The cC is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 6 congruent (and regular) squares, and 12 congruent flattened hexagons.For a certain depth of chamfering, all (final) edges of the chamfered cube have the same length; then, the hexagons are equilateral, but not regular.

The cC is also inaccurately called a truncated rhombic dodecahedron, although that name rather suggests a rhombicuboctahedron.

The DaYan Gem 7 is a twisty puzzle in the shape of a chamfered cube.

[2] In geometry, the chamfered octahedron is a convex polyhedron constructed by truncating the 8 order-3 vertices of the rhombic dodecahedron.

These truncated vertices become congruent equilateral triangles, and the original 12 rhombic faces become congruent flattened hexagons.For a certain depth of truncation, all (final) edges of the cO have the same length; then, the hexagons are equilateral, but not regular.

The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 12 congruent regular pentagons and 30 congruent flattened hexagons.It is constructed as a chamfer of a regular dodecahedron.

The pentagons are reduced in size and new faces, flattened hexagons, are added in place of all the original edges.

For a certain depth of chamfering, all (final) edges of the cD have the same length; then, the hexagons are equilateral, but not regular.

The cD is also inaccurately called a truncated rhombic triacontahedron, although that name rather suggests a rhombicosidodecahedron.

In geometry, the chamfered icosahedron is a convex polyhedron constructed by truncating the 20 order-3 vertices of the rhombic triacontahedron.

The hexagonal faces of the cI can be made equilateral, but not regular, with a certain depth of truncation.

The chamfer operation applied in series creates progressively larger polyhedra with new faces, hexagonal, replacing the edges of the current one.

The truncated tetrahedron looks similar; but its hexagons correspond to the 4 faces, not to the 6 edges, of the yellow tetrahedron, i.e. to the 4 vertices, not to the 6 edges, of the red tetrahedron.
Historical drawings of truncated tetrahedron and slightly chamfered tetrahedron. [ 1 ]
Chamfered cube (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.)
The truncated octahedron looks similar; but its hexagons correspond to the 8 faces, not to the 12 edges, of the octahedron, i.e. to the 8 vertices, not to the 12 edges, of the cube.
Historical drawings of rhombic dodecahedron and slightly chamfered octahedron
The truncated icosahedron looks similar, but its hexagons correspond to the 20 faces, not to the 30 edges, of the icosahedron, i.e. to the 20 vertices, not to the 30 edges, of the dodecahedron.
A chamfered square
(See also the previous version of this figure.)